Dispersion Using SPSS Output Hours watching TV for Soc 3155 students: 1. What is the range & interquartile range? 2. Is there skew (positive or negative) in this distribution? 3. What is the most common number of hours reported? 4. What is the average squared distance that cases deviate from the mean? Statistics Hours watch TV in typical week NValid18Missing11 Mean Median Mode5.00 Std. Deviation Variance Minimum 1.00 Maximum Percentiles

Hypothesis practice For each, write the null hypothesis and indicate whether it is directional Crime rates are related to unemployment rates People with lumpy heads are less likely to shave their hair off There is a difference in criminal behavior between those who live with parents and those who do not Those with prior felony offenses will be more likely to commit new crimes than those without prior offenses

The Normal Curve & Z Scores

THE NORMAL CURVE Characteristics: Theoretical distribution of scores Perfectly symmetrical Bell-shaped Unimodal Tails extend infinitely in both directions x AXIS Y axis

THE NORMAL CURVE Assumption of normality of a given empirical distribution makes it possible to describe this “real-world” distribution based on what we know about the (theoretical) normal curve

THE NORMAL CURVE.68 of area under the curve (.34 on each side of mean) falls within 1 standard deviation (s) of the mean In other words, 68% of cases fall within +/- 1 s 95% of cases fall within 2 s’s 99% of cases fall within 3 s’s

Areas Under the Normal Curve Because the normal curve is symmetrical, we know that 50% of its area falls on either side of the mean. FOR EACH SIDE:  34.13% of scores in distribution are b/t the mean and 1 s from the mean  13.59% of scores are between 1 and 2 s’s from the mean  2.28% of scores are > 2 s’s from the mean

THE NORMAL CURVE Example: Male height = normally distributed, mean = 70 inches, s = 4 inches What is the range of heights that encompasses 99% of the population?  Hint: that’s +/- 3 standard deviations

THE NORMAL CURVE & Z SCORES –To use the normal curve to answer questions, raw scores of a distribution must be transformed into Z scores Z scores: Formula: Z i = X i – X s –A tool to help determine how a given score measures up to the whole distribution RAW SCORES: Z SCORES:

NORMAL CURVE & Z SCORES Transforming raw scores to Z scores a.k.a. “standardizing” converts all values of variables to a new scale:  mean = 0  standard deviation = 1 Converting raw scores to Z scores makes it easy to compare 2+ variables Z scores also allow us to find areas under the theoretical normal curve

Z SCORE FORMULA Z = X i – X S X i = 120; X = 100; s=10 –Z= 120 – 100 = X i = 80, S = 10 X i = 112, S = 10 X i = 95; X = 86; s=7 Z= 80 – 100 = Z = 112 – 100 = Z= 95 – 86 =

USING Z SCORES FOR COMPARISONS –Example 1: An outdoor magazine does an analysis that assigns separate scores for states’ “quality of hunting” (MN = 81) & “quality of fishing” (MN =74). Based on the following information, which score is higher relative to other states? Formula: Z i = X i – X s –Quality of hunting for all states: X = 69, s = 8 –Quality of fishing for all states: X = 65, s = 5

USING Z SCORES FOR COMPARISONS –Example 2: You score 80 on a Sociology exam & 68 on a Philosophy exam. On which test did you do better relative to other students in each class? Formula: Z i = X i – X s –Sociology: X = 83, s = 10 –Philosophy: X = 62, s = 6

Normal curve table For any standardized normal distribution, Appendix A (p ) of Healey provides precise info on: the area between the mean and the Z score (column b) the area beyond Z (column c) Table reports absolute values of Z scores Can be used to find: The total area above or below a Z score The total area between 2 Z scores

THE NORMAL DISTRIBUTION Area above or below a Z score If we know how many S.D.s away from the mean a score is, assuming a normal distribution, we know what % of scores falls above or below that score This info can be used to calculate percentiles

AREA BELOW Z EXAMPLE 1: You get a 58 on a Sociology test. You learn that the mean score was 50 and the S.D. was 10. –What % of scores was below yours? Z i = X i – X = 58 – 50 = 0.8 s 10

AREA BELOW Z What % of scores was below yours? Z i = X i – X = 58 – 50 = 0.8 s 10 Appendix A, Column B (28.81%) of area of normal curve falls between mean and a Z score of 0.8 Because your score (58) > the mean (50), remember to add.50 (50%) to the above value.50 (area below mean) (area b/t mean & Z score) =.7881 (78.81% of scores were below yours) YOUR SCORE WAS IN THE 79 TH PERCENTILE FIND THIS AREA FROM COLUMN B

AREA BELOW Z –Example 2: –Your friend gets a 44 (mean = 50 & s=10) on the same test –What % of scores was below his?

AREA BELOW Z What % of scores was below his? Z = X i – X = 44 – 50= -0.6 s 10 FIND THIS AREA FROM COLUMN C

Z SCORES: “ABOVE” EXAMPLE –Sometimes, lower is better… Example: If you shot a 68 in golf (mean=73.5, s = 4), how many scores are above yours? 68 – 73.5 = FIND THIS AREA FROM COLUMN B

Area between 2 Z Scores What percentage of people have I.Q. scores between Stan’s score of 110 and Shelly’s score of 125? (mean = 100, s = 15) CALCULATE Z SCORES

AREA BETWEEN 2 Z SCORES EXAMPLE 2: The mean prison admission rate for U.S. counties is 385 per 100k, with a standard deviation of 151 (approx. normal distribution)  Given this information, what percentage of counties fall between counties A (220 per 100k) & B (450 per 100k)?

4 More Sample Problems For a sample of 150 U.S. cities, the mean poverty rate (per 100) is 12.5 with a standard deviation of 4.0. The distribution is approximately normal. Based on the above information: 1. What percent of cities had a poverty rate of more than 8.5 per 100? 2. What percent of cities had a rate between 13.0 and 16.5? 3. What percent of cities had a rate between 10.5 and 14.3? 4. What percent of cities had a rate between 8.5 and 10.5?